112 THE ROYAL SOCIETY OF CANADA 
§ 11. As reference has been made in §10 to the integral sine 
function, it may not be out of place here to correct a misprint in the 
formula (3) for S; (2x). (N. p. 295). The numerator 8 should be 4. 
The corrected formula 
S, (2x) =ax [3/7 Le 2 oo my Sosy Ji 17 (27) 
can be made to A at once upon es aati of § 10. 
Replacing —— i by 2 2 S = =) , the expression on the right 
rt ieee See er) 
hand side of (27) can be written 
TX [Ja (7: dns 3) +3 Ja Wits )\+3J5 J, TJ; DT. 
= TX BE 
2 
=T Dai ts eis Soe +. ..]=S, (2x) 
Thus (27) is a RER ee of (24). 
§ 12. Thæidentity 
2s+1 2s+1 2s+1 ee 
shows that 
Antero SEL DO 
25+ 1 5 2s+1 
Cts 10 
1 AE 1) 2s+1 
‘oT GED!” 
=1 Z, (2x) (29) 
where Z, (x) is Siemon’s function, defined in $ 6. 
Another expansion for Zp (2x) can be derived from the formula 
= EU This AI g4. ey 

Zy (2%) = = cE sin (2x cos ¢)d@ 
= I; sin (2x sin @) dp 
[ GS sino 1 Gx) Hour 
[Ji (2x) +3 Js (2x) +5 Js (2x) +. ..] (30) 
